Embedded newform 224.3.w.a.43.14

• Label: 224.3.w.a.43.14
• Level: $224$
• Weight: $3$
• Character: 224.43
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			43.14
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34412 - 1.48100i) q^{2} +(1.15434 + 2.78683i) q^{3} +(-0.386709 + 3.98126i) q^{4} +(-3.15510 + 7.61708i) q^{5} +(2.57571 - 5.45539i) q^{6} +(1.87083 - 1.87083i) q^{7} +(6.41602 - 4.77856i) q^{8} +(-0.0699303 + 0.0699303i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34412	−	1.48100i	−0.672058	−	0.740499i
							\(3\)	1.15434	+	2.78683i	0.384780	+	0.928942i	0.991027	+	0.133665i	\(0.0426745\pi\)
−0.606246	+	0.795277i	\(0.707325\pi\)
\(5\)	−3.15510	+	7.61708i	−0.631020	+	1.52342i	0.207323	+	0.978273i	\(0.433525\pi\)
							−0.838343	+	0.545144i	\(0.816475\pi\)
\(7\)	1.87083	−	1.87083i	0.267261	−	0.267261i
							\(11\)	−3.33495	+	8.05128i	−0.303177	+	0.731935i	0.696716	+	0.717347i	\(0.254644\pi\)
−0.999894	+	0.0145881i	\(0.995356\pi\)
\(13\)	−2.60255	−	6.28312i	−0.200196	−	0.483317i	0.791616	−	0.611019i	\(-0.209240\pi\)
							−0.991813	+	0.127702i	\(0.959240\pi\)
\(17\)			13.1664i			0.774494i	0.921976	+	0.387247i	\(0.126574\pi\)
							−0.921976	+	0.387247i	\(0.873426\pi\)
\(19\)	−26.6062	+	11.0207i	−1.40033	+	0.580035i	−0.949837	−	0.312746i	\(-0.898751\pi\)
							−0.450492	+	0.892781i	\(0.648751\pi\)
\(23\)	−7.99211	−	7.99211i	−0.347483	−	0.347483i	0.511688	−	0.859171i	\(-0.329020\pi\)
							−0.859171	+	0.511688i	\(0.829020\pi\)
\(29\)	−5.63830	+	2.33546i	−0.194424	+	0.0805330i	−0.477771	−	0.878484i	\(-0.658555\pi\)
							0.283347	+	0.959017i	\(0.408555\pi\)
\(31\)		−	39.2305i		−	1.26550i	−0.774357	−	0.632749i	\(-0.781926\pi\)
							0.774357	−	0.632749i	\(-0.218074\pi\)
\(37\)	−19.0910	+	46.0898i	−0.515973	+	1.24567i	0.424385	+	0.905482i	\(0.360490\pi\)
							−0.940358	+	0.340187i	\(0.889510\pi\)
\(41\)	−19.5420	+	19.5420i	−0.476634	+	0.476634i	−0.904053	−	0.427420i	\(-0.859423\pi\)
							0.427420	+	0.904053i	\(0.359423\pi\)
\(43\)	−4.86423	+	11.7433i	−0.113122	+	0.273100i	−0.970294	−	0.241931i	\(-0.922219\pi\)
							0.857172	+	0.515030i	\(0.172219\pi\)
\(47\)	89.7695			1.90999			0.954995	−	0.296623i	\(-0.0958605\pi\)
							0.954995	+	0.296623i	\(0.0958605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.14	✓	192
32.3	odd	8	inner	224.3.w.a.99.14	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.14	✓	192	1.1	even	1	trivial
224.3.w.a.99.14	yes	192	32.3	odd	8	inner